Classical Langevin Processes¶

The homodyne framework is built on a Gaussian model for particle dynamics, motivated by the classical Langevin equation. This page derives the transport coefficient \(J(t)\) for the four canonical stochastic processes relevant to soft-matter XPCS experiments, and compares their physical behaviour.

The Langevin Equation¶

The overdamped Langevin equation for a colloidal particle in one dimension is:

(1)¶\[m\ddot{x}(t) \;=\; -m\gamma\,\dot{x}(t) + \eta(t) + F_\mathrm{ex}(t)\]

where:

\(m\) is the particle mass
\(\gamma\) is the friction coefficient (\(\text{s}^{-1}\))
\(\eta(t)\) is Gaussian white noise with \(\langle\eta(t)\rangle = 0\) and \(\langle\eta(t)\eta(t')\rangle = 2m^2\gamma k_B T\,\delta(t-t')\) (fluctuation-dissipation theorem)
\(F_\mathrm{ex}(t)\) is an external deterministic force (shear, applied field, etc.)

For colloidal particles (large \(m\gamma \tau_B \gg 1\), where \(\tau_B = m/\gamma\) is the Brownian relaxation time), the inertial term \(m\ddot{x}\) is negligible on experimental timescales, giving the overdamped equation:

(2)¶\[\dot{x}(t) \;=\; \mu\,F_\mathrm{ex}(t) + \sqrt{2D}\,\xi(t)\]

where \(\mu = 1/(m\gamma)\) is the mobility, \(D = k_B T\mu\) (Einstein relation), and \(\xi(t)\) is unit-variance white noise.

Wiener Process (Standard Diffusion)¶

Setting \(F_\mathrm{ex} = 0\) in (2), the position evolves as a Wiener process (Brownian motion):

\[x(t) \;=\; x(0) + \sqrt{2D}\int_0^t \xi(t')\,dt'\]

Variance (MSD):

\[\mathrm{Var}[x(t) - x(0)] \;=\; 2Dt\]

Transport coefficient:

(3)¶\[J(t) \;=\; \frac{d}{dt}\,\mathrm{Var}[x(t)] \;=\; 2D \;=\; \text{const.}\]

Equilibrium \(g_2\): simple exponential decay with \(\Gamma = Dq^2\) and \(g_2(q,\tau) = 1 + \beta\,e^{-2Dq^2\tau}\).

XPCS application: This is the baseline model for Brownian diffusion in dilute suspension. The Stokes-Einstein relation gives \(D = k_BT/(6\pi\eta R)\) for a sphere of radius \(R\) in a medium of viscosity \(\eta\).

Ornstein-Uhlenbeck Process (Inertial Brownian Motion)¶

When inertial effects are retained (finite particle mass \(m\)), the full Langevin equation (1) with \(F_\mathrm{ex} = 0\) reads:

(4)¶\[m\ddot{x}(t) \;=\; -m\gamma\,\dot{x}(t) + \eta(t)\]

The velocity \(v(t) = \dot{x}(t)\) follows an Ornstein-Uhlenbeck process:

\[\dot{v}(t) \;=\; -\gamma\,v(t) + \tfrac{1}{m}\,\eta(t)\]

with relaxation rate \(\gamma\) (the friction coefficient from (1)). The momentum relaxation time is \(\tau_B = 1/\gamma\).

Note

The velocity (not the position) is the OU variable here. The position \(x(t) = \int_0^t v(t')\,dt'\) is the integral of the OU velocity, and its variance grows without bound (free diffusion at long times). A position-OU process — where a harmonic potential confines the particle — gives qualitatively different behaviour; see the Brownian oscillator sections below.

Velocity covariance (starting from \(v(0) = 0\)):

\[\mathrm{Cov}\!\left[v(t),\, v(t')\right] \;=\; D\gamma\!\left(e^{-\gamma|t-t'|} - e^{-\gamma(t+t')}\right)\]

Transport coefficient (from the Green-Kubo relation (3)):

(5)¶\[J(t) \;=\; 2D\!\left(1 - e^{-\gamma t}\right)^2\]

Physical behaviour:

At short times (\(t \ll 1/\gamma\)): \(J \approx 2D\gamma^2 t^2\) (ballistic regime — inertia dominates and displacement variance grows as \(t^3\)).
At long times (\(t \gg 1/\gamma\)): \(J \to 2D\) (diffusive regime — velocity correlations have fully decorrelated, recovering free Brownian motion).
The crossover occurs at the momentum relaxation time \(\tau_B = 1/\gamma\).

XPCS application: For colloidal particles in water, \(\tau_B \sim 10^{-9}\) s (far below XPCS time resolution), so the OU correction is negligible. It becomes relevant for nanoparticles, proteins, or ultrafast XPCS at next-generation sources. In the overdamped limit (\(\gamma \to \infty\) with \(D\) fixed), \(J \to 2D\) instantly, recovering the Wiener process. Conversely, the overdamped Brownian oscillator (below) reduces to this OU result in the limit \(\omega_0 \to 0\).

Brownian Oscillator (Underdamped)¶

A particle in a harmonic trap with weak damping (\(\gamma < 2\omega_0\)) exhibits oscillatory dynamics before relaxing:

(6)¶\[m\ddot{x} + m\gamma\dot{x} + m\omega_0^2 x \;=\; \eta(t)\]

Define the shifted frequency \(\omega_s = \sqrt{\omega_0^2 - \gamma^2/4}\).

Transport coefficient (underdamped):

(7)¶\[J(t) \;=\; \frac{2D\gamma^2}{\omega_s^2}\, e^{-\gamma t}\sin^2\!\left(\omega_s t\right)\]

Physical behaviour:

Oscillates at frequency \(\omega_s\) (ringing).
Envelope decays at rate \(\gamma\) (damping).
Maximum \(J_\mathrm{max} = 2D\gamma^2/\omega_s^2\) at \(\tau = \pi/(2\omega_s)\).

XPCS application: Models colloidal particles in strongly viscoelastic media (gels, glasses near the Lindemann criterion) where particles oscillate within cages before escaping.

Brownian Oscillator (Overdamped)¶

When damping is strong (\(\gamma > 2\omega_0\)), define the shifted decay rate \(\gamma_s = \sqrt{\gamma^2 - 4\omega_0^2}\).

Transport coefficient (overdamped):

(8)¶\[J(t) \;=\; \frac{8D\gamma^2}{\gamma_s^2}\, e^{-\gamma t}\sinh^2\!\left(\tfrac{1}{2}\gamma_s t\right)\]

Physical behaviour:

Monotone growth then decay (no oscillations).
At very long times: \(J \to 0\) (particle becomes truly localized if the trap dominates).
Equivalent to superposition of two exponential relaxation modes.

XPCS application: Models particles in dense suspensions or gels where the cage effect completely suppresses oscillation but limits long-time diffusion.

Comparison Table¶

The following table summarizes the four processes:

Process	\(J(t)\)	Long-time limit	Physical regime
Wiener	\(2D\)	\(2D\) (free diffusion)	Dilute suspension
Ornstein-Uhlenbeck	\(2D\left(1-e^{-\gamma\tau}\right)^2\)	\(2D\) (free diffusion)	Inertial effects (\(t \sim \tau_B\))
Brown. Osc. (underdamped)	\(\frac{2D\gamma^2}{\omega_s^2}e^{-\gamma\tau}\sin^2(\omega_s\tau)\)	0 (localized)	Colloidal crystal, gel
Brown. Osc. (overdamped)	\(\frac{8D\gamma^2}{\gamma_s^2}e^{-\gamma\tau}\sinh^2(\frac{\gamma_s\tau}{2})\)	0 (localized)	Dense glass, arrested gel

Variance of displacement \(\mathrm{Var}[x(t_2)-x(t_1)] = \int_{t_1}^{t_2} J(t')dt'\):

Process	\(\mathrm{Var}[x(t_2)-x(t_1)]\)
Wiener	\(2D\tau\)
OU	\(2D\!\left[\tau - \frac{2}{\gamma}(1-e^{-\gamma\tau}) + \frac{1}{2\gamma}(1-e^{-2\gamma\tau})\right]\)
Brown. Osc. (underdamped)	\(\frac{D\gamma^2}{\omega_s^2\omega_0^2}\left[2\omega_0^2\tau - 2\gamma(1-e^{-\gamma\tau}) + \frac{\gamma^2+\omega_s^2}{\omega_s}\sin(2\omega_s\tau)e^{-\gamma\tau} + \cdots\right]\)
Brown. Osc. (overdamped)	Analogous expression with \(\sinh\) terms

Non-Gaussian Corrections¶

The Langevin framework predicts Gaussian displacement statistics. Deviations — measured by the non-Gaussian parameter:

\[\alpha_2(t) \;=\; \frac{\langle\Delta x^4\rangle}{3\langle\Delta x^2\rangle^2} - 1\]

— signal heterogeneous dynamics, multi-population behaviour, or strongly anharmonic potentials. For colloidal suspensions near the yielding transition (He et al. PNAS 2025):

Repulsive suspensions: \(\alpha_2 \approx 0\) (Gaussian, homodyne model valid).
Attractive suspensions: \(\alpha_2 \gg 0\) (power-law tails, heterodyne model needed).

This non-Gaussian criterion determines whether the homodyne or heterodyne formalism should be applied to a given experiment.